2D frames optimization. Criterion: maximum stability

- Collapse load and mechanism of steel frames by slender bars with variable section.
- Plastic design allows more rational solutions and remarkable economy of materials.
- 2D element bar is formulated with located plastic hinges modelled by semirigid knots.
- The method is novel in that it is based on the differential bar formulation.
- It is not necessary to calculate the matrix rigidity neither the equivalent forces.

Structural analysis was one of the first disciplines to demand powerful computing tools. However, with current capacities of both calculus and manufacture and use of new materials, along with certain aesthetic conditions, it is possible to address problems such as the one presented in this article, whose aim is the optimal variation of any frame, so that few criteria are met, including stability. The problem is complex and must be solved numerically. This paper presents a formulation for solving the optimization problem, considering not only the buckling conditions but any other, such as allowable stress or limited displacement. The equilibrium of each beam in its deformed geometry is proposed under assumption of small displacements and deformations (Second Order Theory). The optimization problem is mathematically formulated to determine which values maximize the buckling load of the frame and numerically solved by sequential quadratic programming. Finally, for the optimal solution from the point of view of stability, the plastic collapse load is calculated. The plastic behavior is based on the bending moment and leads to sudden concentrated plastic hinges. Therefore, the structural stability is affected, which is checked during the loading process.